Sea quark contribution to the dynamical mass and light quark content of a nucleon

PHYSICAL REVIEW D VOLUME 5 1, NUMBER 1 1 1 JUNE 1995

Sea quark contribution to the dynamical mass and light quark content of a nucleon

Janardan P. Singh School of Studies in Physics, Jiwaji University, Gwalior -474011, India

(Received 4 March 1994)

We calculate the flavor mixing in the wave function of a light valence quark. For this, we use the idea of dynamical symmetry breaking. A sea quark of a different flavor may appear through the vacuum polarization of a gluon propagator which appears in the gap equation for the dynamical mass. We have also used the fact that any one of these quark lines may undergo condensation. The dependence of the dynamical mass, generated in this way, on the sea quark mass up to quadratic terms has been retained. The momentum dependence is like llp4, in contrast with the llpZ kind of dependence which occurs for the leading term of the dynamical mass in the subasymptotic region. The extension of the result to the "mass shell" yields a , ~ = 53-54 MeV for the pion-nucleon a term and rn,(plsslp) = 122-264 MeV for the strange quark contribution to the proton mass, for different values of parameters. These are in reasonable agreement with current phenomenological estimates of these quantities.

PACS number(s): 12.39.-x, 12.38.Lg, 14.20.Dh, 14.65.Bt

I. I N T R O D U C T I O N

Recent analysis and observation of flavor mixing in the wave function of the valence quarks in the baryon sector cast doubt on our understanding of the constituent quark model and the Okubo-Zweig-Iizuka (OZI) rule. Thus the determination of the C term (CTN) , from pion-nucleon scattering and its implications for the strangeness in the proton [I-71, the quark-spin content of the proton, i.e., the flavor singlet axial-charge g; of the nucleon [8], and intrinsic charm quarks in the nucleon suggested by J / $ , and open-charm production in hadron-nucleus collisions [9], do not go well with the naive interpretation of the constituent uuark model.

From deep inelastic scattering experiments, it is known that the nucleon consists of not only valence quarks but essentially a n infinite number of quark-antiquark pairs as well. The interpretation of a constituent quark as a valence quark and many qq pairs and gluons is as old as QCD itself. IVevertheless, it is a well known fact that for low energy phenomena, such as the magnetic moments of baryons, the spectroscopy of mesons and baryons, the meson baryon couplings, and ratios of total cross sec- tions such as u ( r N ) / u ( N N ) , etc., the constituent quark model works remarkably well. I t is not unreasonable to assume that for quasistatic processes such as g; of the nucleon and more, especially for U=N, which is determined from essentially low energy pion-nucleon scattering experiments, the constituent quark picture should not be very much off the mark.

The experimental determination of the C term comes from the investigation of rip -t rkp scattering. Specifi- cally one takes the isospin-even scattering amplitude with the Born term subtracted (called D+ in the literature). The pion-nucleon C term is defined as

where F, is the pion decay constant and the kinematic location u = 0, t = 2M: is called the Cheng-Dashen (CD) point. Since the CD point is outside the physical region, one needs to extrapolate the available experimental information to this point using analyticity properties and dispersion theory. The Karlsruhe group's value is generally the standard [lo], with

CTN = 64 i 8 MeV

In order to do the theoretical evaluations of the C term, one first defines another related quantity U,N as the nucleon matrix element of the double commutator of the symmetry-breaking term in the baryon Hamiltonian with two axial charges, where an average over proton and neutron states is understood:

A formal evaluation of the commutator in QCD yields

The chiral symmetry relates the value of the CTN to the nucleon matrix element of the quark mass term [ l l ] :

The fact that u , ~ = u(O), thus relates CTN to u , ~ :

where A, = u ( 2 i E ) - u(0) and is the quantity of crucial interest; AR is of the order of M,4 1n A42 and is numerically small: AR = 0.35 MeV. Using the quark mass expansion of the energy levels in the baryon octet, one

6338 @ 1995 The American Physical Society

51 - SEA QUARK CONTRIBUTION TO THE DYNAMICAL MASS A N D . . . 6339

can, on the other hand, express o , ~ in terms of physical contributions of sea quarks as well. The important ques- masses and an additional parameter y, defined by tion here is the following: To what extent can a picture

of a constituent quark as a valence quark dressed with 2(N(sslN) sea quarks and gluons be derived from the basic laws of

Y = (N~(G ,u + dd)lN) QCD? Being a low-energy phenomena, it is deeply related

to the non~erturbative aspects of QCD. In the present which reflects the strange quark content of the proton. article, we have tried to understand the flavor mixing in At the one-loop level in chiral perturbation theory, the the valence quark using the idea of dynamical symmetry expression for U,N becomes [3] breaking and have extended it to estimate the light quark

scalar condensate between one nucleon states. A reliable 6

U,N = - , 6 = (35 f 5) MeV, (1.5) theoretical determination of such low-energy parameters 1 - Y using the basic laws of QCD is important.

where the 5 MeV error is due to the theoretical uncer- tainty in the terms of the order O(mg).

At the one-loop level in chiral perturbation theory, Gasser, Sainio, and Svarc [4] have found that A, = 4.6 MeV resulting in

C,N - U,N 5 MeV. (1.6)

However, a dispersion analysis gives a higher result [5], A, = 15.2 f 0.4 MeV, and this yields

U,N N 49 i 8 MeV. [I.7)

In another method based on forward dispersion re- lations, the authors [6] have extracted CTN from low- energy data. In this procedure C,N is parametrized as a function of the scattering lengths, Gasser, Leutwyler, and Sainio [7] have used this to obtain

f f , ~ g 4 5 f 4 f 4 f 4 MeV, (I.a)

where errors refer to statistics, to database modifica- tions, and to the input uncertainties, respectively. They have confirmed the old estimate CTN N 60 MeV also. The rapid increase in u(t) from o(0) % 45 MeV to a(2M:) " 60 MeV is attributed to the anomalous thresh- old associated with mr intermediate states.

The difference in numerical values of 6 obtained from mass splittings of octet baryons [Eq. (1.5)], and a , ~ deduced from pion-nucleon scattering data is interpreted as a manifestation of nontrivial matrix element of scalar strange quark operator between one nucleon states. Its contribution to the nucleon mass term is believed to be larger than o , ~ [1,2,7]:

m,(NlssjN) 130 MeV,

where only A, 15 MeV has been used, since the one- loop result (1.6) is afflicted by large higher-order corrections 171.

L ,

Calculations of the o term ( U , ~ ) have been attempted by various authors in various ways: the chiral soliton model [12], linear o model [13], Nambu-Jona-Lasinio model [14], bag model [12], QCD sum rule approach [15], etc. Our point of view in this article will be that the method of reconciling theoretical and experimental estimates of u terms along with the violations of the OZI rule in the form of Eq. (1.9) requires the construction of a constituent quark picture that takes into account

1t is well known that light quarks are attributed to two kinds of masses. The current quark masses are small, typ- ically 5-10 MeV for u and d quarks, and are important for current algebra calculations and for deep inelastic scattering processes. The dynamical masses are relatively large-these are - 300 MeV for u and d quarks-and - are important to understanding hadron spectroscopy, nucleon magnetic moments, etc. The constituent masses have to be understood as a sum of these two kinds of masses, evaluated at the constituent mass scale [16]. On the other hand, according to the well known Feynman- Hellman theorem, the matrix elements of scalar quark currents between one nucleon states can be derived as

where mN is the nucleon mass and mi is the current quark mass of the quark in question. In the constituent quark picture, the hadron mass is supposed to be made up of constituent quark masses. If the phenomenological estimate (1.9) has any truth to it, it means that the sea quarks, and, in particular, the strange quarks, have a role in the constitution of the constituent quark mass or the dynamical mass, for that matter. In the process of forming a constituent quark, the quark is "dressed" by gluonic and even 3s quark fields. It is no longer the naive object that occurs in the QCD Lagrangian. It is this dressed object which may generate strange quark matrix elements [2]. Here, the strange quark may have a special role due to the proximity of its mass to the QCD scale parameter:

The above point of view [16] is not the only one which connects the constituent quark model (CQM) to QCD. Damgaard, Nielsen, and Sollacher [17] have used a gauge symmetric approach for extracting effective degrees of freedom from QCD and have identified chirally rotated quark fields with constituent quarks. A related and perhaps complementary picture has been suggested by Kaplan [18], in which constituent quarks are viewed as Skyrmions in color space. Wilson et al. [19] have argued that CQM and QCD can be reconciled in light front dy- namics.

In the usual approaches to dynamical symmetry breaking, it is the interaction of the particular (light) quark with gluons which is responsible for the generation of the dynamical mass of the quark [20]. In such an approach, the dynamical mass (evaluated on "mass shell")

JANARDAN P. SINGH - 5 1

apart from some calculable numerical factors [21], since AQcD is the only mass scale available in the theory. How- ever, if we consider a theory which has other mass parameters, such as current quark masses, there is no reason why these additional mass parameters will not contribute to the generation of the dynamical mass.

To simplify the discussion, let us consider QCD with only three flavors of light quarks. The heavier quarks may be taken as decoupled in a n effective way in the manner of the Appelquist-Carazzone theorem [2]. Among the light quarks, we may consider the masses of the u and d quarks (m, << A 4 c ~ ) to be zero, leaving only m, as the other nontrivial mass parameter. In such a theory, the dynamical mass and the quantities such as quark and gluon condensate, which were earlier expressible in terms of AQcD only [22], should now involve m, as well. For instance, the light quark condensate can now be written as

where A = AQcD and a , b, c are numerical coefficients, which may include logarithms in mass parameters, and are presumably in decreasing order. For instance, the leading chiral behavior of the quark condensate in terms of m z (or equivalently mq, since m i cx mq) as an expansion parameter has been worked out by Novikov et al. [23]; Leutwyler and Smilga [24] have tried an expansion of the quark condensate in terms of the current quark mass. Obviouslv this dependence should be different de- pending on whether the current quark mass pertains to the same quark or a quark of different flavor. I t is this kind of dependence which leads to isospin [25] and SU(3) splitting of quark condensates which otherwise would be the same for all the light quarks in the chiral limit.

The simplest way in which the dependence of C(p2)d,, on the current quark mass of a sea quark may be incorporated is through the vacuum polarization of a gluon line that appears in the Schwinger-Dyson equation for the mass function of the quark. In this connection, recently [26] it has been emphasized that in QCD the vacuum polarization can play an important role in the physics of light quarks. We can improve this simple approach by invoking operator product expansion and supplementing this diagram with the ones which have nonperturbative vacuum condensates, which appear in QCD sum rule methods as phenomenological constants [27]. In a sys- tematic approach, all these contributions should be down compared to the leading contribution to the dynamical mass.

In Sec. 11, we derive a dependence of the dynamical mass function on the current quark mass of a light sea quark by incorporating the vacuum polarization in the gluon propagator, which will be used for writing the Schwinger-Dyson equation. In Sec. 111, we extend our result to compute the pion-nucleon C term and the strange quark content of the nucleon. In Sec. IV, we discuss the results and give our conclusions. Finally, we give some mathematical details in the appendices.

11. FLAVOR MIXING IN DYNAMICAL MASS APPROACH

We start with a QCD with three light flavors of quarks u, d, and s. We take m, md << m, and m, N AqcD. The heavier quarks may be considered as decoupled in an effective way, their contribution being O ( m h 2 ) through a gluon vacuum polarization [2]. The dynamical mass of the light quarks has been calculated mainly using two methods: (i) by solving the gap equation obtained by writing the Schwinger-Dyson equation for the quark propagator [20] and (ii) by using the operator product expansion (OPE) of those nonperturbative vacuum expectation values which are known to contribute to QCD sum- rule phenomenology [16]. In either approach, the asymptotic behavior of the dynamical mass function comes out to be

apart from a logarithm, which, in any case, we will ne- glect in our discussion for the sake of simplicity. In fact, for our purpose, we shall use the following convenient parametrization of the dynamical mass function [28]:

I t has the property that a t p2 = M2, C D = M and the asymptotic behavior for large p2 is retained. I t can be extended down to the low-.energy region as well, a t least when it appears within integration. We first compute the self-energy of a u or d quark for the diagram corresponding to Fig. 1, where the hatched area denotes the quark propagator with the dynamical mass (2.1) incorporated, but the current quark mass neglected. A sea quark con- tributes through the vacuum polarization of the gluon propagator. For a given circulating quark of mass ,mi, we use [29]

- i p b k,ku g2 2 m2 (3z2 - 2z3)(1 - 2z)k2 ~ ; ; ( l ) (k) = -

k2 + i~ (gw - ~ ~ ( - 1 ) [ - ln+ + 1 kzZ(l - z ) - m? I for the gluon propagator with vacuum polarization. p is the renormalization point and we have written only the mi-dependent part of the expression. We assume that the mass parameter mi appearing in the loop is the current quark mass. The evaluation of diagram ( I ) in Euclidean variables yields

SEA QUARK CONTRIBUTION TO THE DYNAMICAL MASS AND . . .

FIG. 1. Sea quark contribution to the dynamical mass in FIG. 2. Sea quark contribution to the dynamical mass with two loops order. Hatched area denotes the dynamical mass condensing of the valence quark lines. of the valence quark.

where Co = CD given in Eq. (2.1). In order to make the one replaces C2 in the denominator by M 2 [30], we get a integration analytically doable, we set the arguments of somewhat different result: the coupling to be p2. This is consistent with our earlier approximation wherein we have neglected the logarithm 6 ~ b ' ) ( ~ 2 ) = .a_ a2(p2) _ M 3 - 3 _ M2 m2 [ 1 . 2 appearing in ED. Obviously this approximation is not 3.rr2 pz 2 p2 p2 going to alter the power behavior of our results below [Eqs. (2.4) and (2.4')] for 6 C ~ ' ( p 2 ) . In Sec. 111, where m: we have done a numerical evaluation of miaC/arni, we -& (18 + 2 in - + 6 ln

have restored the variable nature of the coupling in the p2 P2

- - integrand. Even now the integration is not so easy. We have given the details of this integration in Appendix We shall supplement 6 C r ) with the nonperturbative

A. In the widely used linear approximation [20] where contributions that appear as a consequence of the OPE of nonperturbative vacuum expectation values. Figures 2 C2 in the denominator is set to zero, the result of the and 3 show the lowest nontrivial contributions where calculation. UD to the lowest order in 1/v2. is

, I , - open fermion lines represent the quark condensates. Fur- thermore, we improve the constant field approximation for the vacuum quarks by taking the value of the parameter (D is the covariant derivative)

that specifies the average virtuality of the vacuum quarks Here we have confined ourselves to only those terms to be nonzero. More specifically, in Minkowski space, we which involve mi. In a different approximation, where have chosen

1

where a, C are color indices and j , n are Dirac indices and it is understood that (y - z)' < 0. The expansion in the mass parameter (taken to be current quark mass) uses equations of motion [16] whereas the exponential function parametrizes the contributions of higher-order operators [31].

In the lowest order of llp2, the parameter X i does not contribute to the diagram corresponding to Fig. 2. The diagram corresponding to Fig. 3 diverges in the in-

I

frared region due to the appearance of higher powers of momenta in the denominator in the loop integral. As usually done in QCD sum rule methods [32], we regulate it by using an infrared cutoff A,, which we choose to be AQcD In the next section, we show numerically that the results, for the lighter quarks, are rather insensitive to the choice of A,. Results of calculations of diagrams corresponding to Figs. 2 and 3 give

JANARDAN P. SINGH - 5 1

FIG. 3. Sea quark cont.ribution to the dynamical mass with condensing of one of the sea quark lines.

Ei in Eq. (2.7) is the exponential integral function. In (2.6) and (2.7) also, we have retained only mi-dependent terms and only magnitudes of quark condensates appear as used in Ref. [33] and shown by Barducci et al. [20].

The dependence of S C ~ ) on A, could have been weak- ened overall by one power of A,, had we used the results of Ref. 1341 for the two-vector current correlation function, which sums all the powers in ma. Unfortunately, the momentum integration, in that case, becomes analytically undoable and it becomes difficult to extract the asymptotic behavior of S C ~ ) . The gluon condensate and other higher-dimensional condensates do not contribute to the lowest-order term in l /p2 considered here.

6 C r ) is a full solution corresponding to the Feynman diagram shown in Fig. 2, since it involves a trivial loop integration [barring the fermion loop integral given in

(2.2)]. However, 6 C r ) and S C ~ ) are not the complete terms corresponding to the respective diagrams, since

the complete term would correspond to the solution of an integral equation. We can find a better solution as follows.

The master equation for the dynamical mass function, say in the linear approximation, is

where D is the scalar function in the full gluon propagator:

Denoting the mi-dependent part of C corresponding to Figs. 1 and 3 as 6C and that of D as -T, we can write

where the second term on the right-hand side (RHS) is

6Co = S C ~ ) + S C ~ ) . We convert the above nonhomoge- neous integral equation in the unknown function 6C into a differential equation, as is usually done in dynamical mass calculations [20], and then solve it in the leading log approximation. Writing 6Co(p2) as

mP A + B l n - + C ln-

p2

we get, for the difference 6C1 = 6C - 6x0,

where 00 = 11 - 2nf/3 and A = AgcD Combining expressions (2.4), (2.6), (2.7), and (2.11), we obtain the total m,-dependent contribution to the dynamical mass function in the subasymptotic limit in the leading order

51 - SEA QUARK CONTRIBUTION TO THE DYNAMICAL MASS AND . . . 6343

Figure 1 has two loops but no operators, whereas Figs. 2 and 3 have only one loop and an operator inserted. This results in increase in powers of p2 in the denominator, apart from an increase in powers of a,, compared to the corresponding cases with one loop lesser, in the asymptotic limit. By the same token, we expect that any loop correction and/or operator insertion to these diagrams will result in an increase in powers of llp2 as well as a,. Incidentally, this also indicates that the usual deriva- tion of CD llp2 is stable under loop corrections for asymptotic momenta. All the calculations, here, have been performed in the Landau gauge which has certain advantages for such calculations (Haymaker in 1201) although it may happen that on the mass shell the gauge dependence drops out [16]. We have not tried to improve our results using renormalization group equations.

At this point it should be added that our normaliza- tion condition for the mass function Co(p2 = M 2 ) = M should be augmented to accommodate new terms as well. For this, let us define C D ( ~ ' ) = Co(p2)+6C(')(p2), where 6C(')(p2) is the contribution to the dynamical mass coming from the complete one-loop corrections and the lowest-dimensional nontrivial operator insertions, which includes (2.12). We should define a new mass scale M' such that CD(p2 = MI2) = MI, then M' is the new generated mass scale which incorporates the effect of the light current quark masses in addition to that of AQcD, whereas M was expressible in terms of AQcD only.

Using Eq. (2.12), we can find out the contribution of a light sea quark qi in the constitution of the dynamical mass of a u or d quark:

dCD/dmi also describes the mixing of a quark of i th flavor into the wave function of a. (constituent) quark of different flavor [14]. Equations (2.12) and (2.13) are the main derivations of this paper. In the next section, we will try to see its phenomenological implications for light baryons.

111. LIGHT Q U A R K SCALAR C O N T E N T O F A NUCLEON

In order to find the matrix elements of light quark scalar operators between one nucleon states, we shall assume (i) that the constituent quark mass can be written as [16]

and (ii) the results obtained from the three diagrams, considered in the previous section, when appropriately extended down to p2 = m~,,,t Z M 2 should be sufficient for the purpose. Finally we shall use Feynman-Hellman theorem (1.10) to calculate the matrix element. The contribution of the expression (2.6) has been extended down to p2 = M 2 as such. However, for the contributions of Figs. 1 and 3, we take the running coupling constant in- side the integration. It has been assumed that g2(k,p)

has the widely used form [20]

Furthermore, for a , (Q2) , we have used the commonly used hard-freeze form 1351:

(12/27){1/ l n ( Q 2 / A k c ~ ) ) for Q2 > Q;, 7r const r H for Q2 < Qi ,

with H = (12/27){1/ ln(Qi/A&,)). Numerically, for the u and d content of the proton

6~:) is important, whereas for the s content of the pro-

ton 6 C r ) gives the dominant contribution. Momentum integration up to Q; can be done analytically; momentum integration beyond Q i and the entire parametric integration has been done numerically. We have taken mu = 5 MeV, m d = 9 MeV, m, = 180 MeV and ( C U ) = (dd), (3s ) = O.g(fiu). We have varied I (qq) / from (0.224 GeV)3 to (0.256 GeV)3, the values normally used in literature [3,16,27]; at the same time M . the parameter that characterizes the dynamical mass of a quark and is close to the constituent quark mass, has been varied from 0.28 to 0.32 GeV, again a range of values that has been normally used in literature [20,33] for the dynamical mass

6344 JANARDAN P. SINGH - 5 1

in order to reproduce the correct hadronic phenomenology. cr,(Qg)/.rr has been varied &om 0.25 (which has been considered as a critical value for dynamical symmetry breaking to occur by some authors [20]) to 0.26 (which is the freezing value of the coupling obtained in Refs. [35,36]); AQcD has been varied from 280 to 300 MeV for three flavors [35] (however, see the results below for AQcD = 200 MeV and cr,/7~ = 0.28, also) and A; has been taken to be 0.4 G e v 2 [31]. For u and d quarks in the sea, results for mu,d(dCD/dmu,d) lies between (5- 6) MeV, giving rise to a n almost unique result for the u term,

U,N S 53 MeV, (3.4) where we have included the contributions of current quark masses of the valence quark as implied by (3.1) and have averaged over proton and neutron. In all the cases, contributions from the region where momentum is larger than Qo turns out to be numerically insignificant. For the s quark, the value of m, (dCD/dm,) has a larger variation (40.5-67.5) MeV which results in

m, ---- dmN " - 122-203 MeV. 8%

In Ref. [37] another form of the "hard freeze" model of coupling has been used to fit the absolute magnitude of the pion-nucleon cross section which needs AQcD = 0.2 GeV and H = 0.28 [see Eq. (3.3)]. The same frozen coupling has been used successfully in subsequent work on deriving nucleon structure functions from the constituent-quark model [38]. With this set of parameters, our result for U,N remains almost unchanged with

U,N 2 54 MeV. (3.6)

However, the same set of parameters result in significant increase in the strange contribution to the nucleon mass:

m , ( d m ~ / d m ~ ) 264 MeV. (3.7)

To see what happens to the integrals if one lets a , (Q2) keep on running below Qg used above, we note that a , (Q2) increases indefinitely rendering any perturbative calculation ineffective as Q2 -+ A2 beyond which it becomes unphysical. Hence in the?%;wing, by the running of a , (Q2) for small Q 2 we basically mean choosing a different value of H corresponding to the lower values of Q:. The result (3.4) has been obtained using Qo/AQcD=2.35 and 2.43 for H = 0.26 and 0.25, respectively. If we choose Qo/AQcD = 2 and 1.5 corresponding to H = 0.32 and 0.55, then for AQcD = 280 MeV the results for m,,d(6'CD/dm,,d) increases by approximately 8 and 20 MeV, respectively.

Thus, our calculated results for ~ , ~ , ( 3 . 4 ) and (3.6), obtained using certain model behavior of the theory for the low-energy region, have a rather good overlap with what has been deduced from experimental and phenomenological analyses. Our estimate of scalar strange quark condensate in a nucleon has a larger scatter, but the lower values obtained in (3.5) are consistent with the current determination of this quantity [see Eq. (1.9)].

We also observe that the numerical results, obtained a t low Euclidean momentum, are quite stable under the change of various parameters when u and d quarks are appearing in the sea. However, the same is not true when the s quark appears in the sea. This is not unexpected, since we have not included terms of O(ma) in 6CtOt. As a consequence, the extrapolation of the result down to p2 = M2 may introduce a larger error in this case, since ms AQCD.

IV. DISCUSSION A N D CONCLUSION

Dynamical symmetry breaking is by now a well known idea which has been used in QCD to describe phenomena in subasymptotic region. The existence of vacuum condensates of fundamental fields in QCD has been well established through successful calculations of numerous low- and intermediate-energy hadronic phenomena [39] and has been theoretically established through computer simulations. We have combined these two ideas to calculate the contribution of a light sea quark to the constituent mass of a light quark of different flavor. The sea quarks in a constituent quark have been assumed to arise as a result of vacuum polarization of gluons which dress the valence quark. Thus we have employed only QCD based techniques to understand one of the low-energy hadronic phenomena. We apply our result t o two different mass scales (mu,d and m,) which differ by over a n order of magnitude. The fact that in both cases our numer- - ical results come reasonably close to what has been found on experimental and phenomenological grounds gives certain credence to our approach.

ACKNOWLEDGMENTS

The author is thankful to Professor J. Pasupathy for some critical comments, Professor H. S. Mani for a critical reading of the manuscript, and U. P. Verma for help in doing numerical computations. Discussions with Avinash Khare and Y. M. Gupta are also acknowledged. The fi- nancial assistance of DST, New Delhi, is gratefully acknowledged.

APPENDIX A: INTEGRALS APPEARING IN 6 ~ 2 ) ( ~ ~ )

In this appendix, we shall perform the following integration (momenta are Euclidean):

which is required for the evaluation of S C ~ ) . First, concentrate on the second term. For the sake of convenience, we

5 1 - SEA QUARK CONTRIBUTION TO THE DYNAMICAL MASS A N D . . . 6345

shall first perform momentum integration and then x integration. Call (32' - 2x3)( l - 22) = A and x ( l - x) = B . The angular integrations can be readily performed:

Thus, the second part reduces to

Although the whole integral is convergent, the parts of it may have divergencies. So, we shall keep upper and lower limits of k2 integration as A t and E'. Recalling that p2 >> M2 > m 2 and at the end, we have to take the limit e2 -+ 0, At -+ m, we get for the second part of the momentum integration of (A.3):

B(p2 + M 2 / 2 ) - m 2 M ~ B ( ~ ~ B - m2 2 m2 ) ] [In n., + ln(m2 + BA:) - 2 in B -

M~ B + [ p 2 B + m 2 2(p2B + m2)3 ] 2(p2B + m 2 )

where we have used expansion of powers and logarithms, keeping in view of the fact that we will need only leading and next-to-leading order terms in the expansion. Also

Subtracting (A4) from (A5), arranging the terms and leaving those terms which drop after x integration on taking the limit of A:, we obtain

The x integration can he conveniently performed, using the factorization of quadratic expressions in x which appear in denominators and logarithms, and by decomposing denominators through partial fractions. We write below our results of prominent integrals only to leading and next-to-leading order in l /p2 . Some of the integrals written in isolation diverge a t the upper limit. In that case we use 1 - E' as the upper limit:

JANARDAN P. SINGH

A dx- = +In&', 1'-' B

A

1-E ' A p2B - m2 m2 m2 dx - " 5 - l n ~ ' + 2 l n - -12-, 1 B p 2 B + m Z F 3 p2 P2

A B ln(B + m2/n;) -1 ,m2 dz - c= .- 1 B (Bp2 + m 2 ) 2 -

In -, m2p2 p2

A 28 2

dx-in B + - r-- 1 - B ( 7 9

2 8 2

A ( 2 ) 9 6m2

dx-ln B + - r-- 1'-' B + ~ n & ~ l n $ - ; ( l n s ) + _ , l-~' Ap2B - m2 ( ) '9" 12 : ( ;22)2 m2 m2 m2 m2

d x ~ p 2 ~ + m 2 In B + , % - - - - + - In- -In&' In- + 26- -4-ln-,

P2 P2 P2 P2

Ap2B - m2 1'-" dx- In B + - 2--- 2 8 ( 7 ) 9 Bp2B + m2

Other prominent integrals needed for our purpose may be obtained from the above integrals by one or two differen- tiations with respect to p2 and/or m2. When all these results are substituted in (A3) all the divergences cancel out and the remaining expressions are finite in the limit A: -+ co.

APPENDIX B: (qq) PROJECTION OF ONE-LOOP GLUON PROPAGATOR

Here, we shall compute the vacuum polarization of a gluon propagator with one quark line condensing in a nonlocal manner. The gluon propagator up to second order in the coupling constant is given by

Let us call the coefficient of -9'12 in the second term on the RHS of (B l ) as I,":. Using Wick's theorem and assuming that one of the quark lines condense, we can write

where the quark propagator is [40]

As stated earlier, (y - z)' < 0. Using Eqs. (2.5) and (B3) we can evaluate (B2). First change the variables as y - z = Y, y + z = 2, then all the integrations, except those in Y can be performed trivially. The result is

where terms O(m2) have been neglected. The result of Y integrations are

SEA QUARK CONTRIBUTION TO THE DYNAMICAL MASS AND . . . 6347

4in2 - - -

[exp (5) - 11 , ) 1'. =

This gives, for t h e to ta l gluon propagator (Bl) ,

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Sea quark contribution to the dynamical mass and light quark content of a nucleon